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Theorem lvoli3 32851
Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
lvoli3.l  |-  .<_  =  ( le `  K )
lvoli3.j  |-  .\/  =  ( join `  K )
lvoli3.a  |-  A  =  ( Atoms `  K )
lvoli3.p  |-  P  =  ( LPlanes `  K )
lvoli3.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvoli3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )

Proof of Theorem lvoli3
Dummy variables  y 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1009 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  P
)
2 simpl3 1010 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  A
)
3 simpr 462 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  -.  Q  .<_  X )
4 eqidd 2421 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  =  ( X 
.\/  Q ) )
5 breq2 4421 . . . . . 6  |-  ( y  =  X  ->  (
r  .<_  y  <->  r  .<_  X ) )
65notbid 295 . . . . 5  |-  ( y  =  X  ->  ( -.  r  .<_  y  <->  -.  r  .<_  X ) )
7 oveq1 6303 . . . . . 6  |-  ( y  =  X  ->  (
y  .\/  r )  =  ( X  .\/  r ) )
87eqeq2d 2434 . . . . 5  |-  ( y  =  X  ->  (
( X  .\/  Q
)  =  ( y 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  r ) ) )
96, 8anbi12d 715 . . . 4  |-  ( y  =  X  ->  (
( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) )  <-> 
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) ) ) )
10 breq1 4420 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  X  <->  Q  .<_  X ) )
1110notbid 295 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  X  <->  -.  Q  .<_  X ) )
12 oveq2 6304 . . . . . 6  |-  ( r  =  Q  ->  ( X  .\/  r )  =  ( X  .\/  Q
) )
1312eqeq2d 2434 . . . . 5  |-  ( r  =  Q  ->  (
( X  .\/  Q
)  =  ( X 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  Q ) ) )
1411, 13anbi12d 715 . . . 4  |-  ( r  =  Q  ->  (
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) )  <-> 
( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) ) )
159, 14rspc2ev 3190 . . 3  |-  ( ( X  e.  P  /\  Q  e.  A  /\  ( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) )  ->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y  .\/  r
) ) )
161, 2, 3, 4, 15syl112anc 1268 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) ) )
17 simpl1 1008 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  HL )
18 hllat 32638 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1917, 18syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  Lat )
20 eqid 2420 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
21 lvoli3.p . . . . . 6  |-  P  =  ( LPlanes `  K )
2220, 21lplnbase 32808 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
231, 22syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  (
Base `  K )
)
24 lvoli3.a . . . . . 6  |-  A  =  ( Atoms `  K )
2520, 24atbase 32564 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
262, 25syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  (
Base `  K )
)
27 lvoli3.j . . . . 5  |-  .\/  =  ( join `  K )
2820, 27latjcl 16241 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( X  .\/  Q )  e.  ( Base `  K
) )
2919, 23, 26, 28syl3anc 1264 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  ( Base `  K ) )
30 lvoli3.l . . . 4  |-  .<_  =  ( le `  K )
31 lvoli3.v . . . 4  |-  V  =  ( LVols `  K )
3220, 30, 27, 24, 21, 31islvol3 32850 . . 3  |-  ( ( K  e.  HL  /\  ( X  .\/  Q )  e.  ( Base `  K
) )  ->  (
( X  .\/  Q
)  e.  V  <->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y  .\/  r
) ) ) )
3317, 29, 32syl2anc 665 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( ( X 
.\/  Q )  e.  V  <->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) ) ) )
3416, 33mpbird 235 1  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   E.wrex 2774   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15073   lecple 15149   joincjn 16133   Latclat 16235   Atomscatm 32538   HLchlt 32625   LPlanesclpl 32766   LVolsclvol 32767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-lat 16236  df-clat 16298  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-lplanes 32773  df-lvols 32774
This theorem is referenced by:  dalem9  32946  dalem39  32985
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